Characterizing lineations defined on open subsets of projective spaces over ordered division rings
Let \(K,L\) be division rings. Denote by \(P^m(K)\) an \(m\)-dimensional projective space over \(K\) \((m\geq 2)\). A map \(F\) of a subset \(U\) of \(P^m(K)\) into \(P^m(L)\) is called a lineation if \(F\) maps any three collinear points to collinear points. Let \(\mathbf{v}=(v_0,\ldots,v_{m+1})\) (resp. \(\mathbf{v}'=(v'_0,\ldots,v'_{m+1}))\) be a reference frame in \(P^m(K)\) (resp. \(P^m(L))\). If \(\psi: K\to L\) is a place, then for \([x_0,\ldots,x^m]\in P^m(K)\) there is an element \(a\) of \(K\) such that \(ax^0,\ldots,ax^m\) are in \(\mathrm{dom}\, \psi,\) but not all in \(\ker \psi\). A map \(P^m(\psi;\underline v,\underline v'): P^m(K)\to P^m(L)\) defined by \([x^0,\ldots,x^m]\mapsto [\psi (ax^0),\ldots,\psi (ax^m)]\) is a lineation. Theorem: Let \(K\) be an ordered division ring, \(v_0,\ldots,v_m, w\) be points in general position in \(P^m(K)\), and \(U\) be a subset of \(P^m(K)\) such that \(U\) contains the simplex \((v_ 0,\ldots,v_m;w)\). If \(F\) is a lineation of \(U\) into \(P^m(L)\) such that \(F(v_0),\ldots,F(v_m),F(w)\) are in general position in \(P^m(L)\), then there is a place \(\phi: K\to L\) such that \(F\) is the restriction of \(P^m(\phi;\underline v,F(\underline v))\) to \(U\), where \(\mathbf{v}\) (resp. \(F(\mathbf{v}))\) is the reference frame \(v_0,\ldots,v_m,w\) (resp. \(F(v_0),\ldots,F(v_m),F(w))\) in \(P^m(K)\) (resp. \(P^m(L))\). It is also shown that let \(K\) be an archimedean and euclidean field, \(U\) be open in \(P^m(K)\), if \(F: U\to P^m(K)\) is an injective lineation such that \(\mathrm{im}\, F\) is not contained in a line, then for any reference frame \(\mathbf{v}\) in \(U\), F is the restriction of \(P^m(\mathrm{id}_K;\underline v,F(\underline v))\) to \(U\). Examples of the fields admitting only surjective endomorphisms, and of non-surjective endomorphisms of fields are given.
scientific article; zbMATH DE number 3958104